What does it mean to write the following?
Let me clarify what I have in mind. The mathematical meaning of is quite obvious. Since , can only mean that , so that if and are real numbers, at least one of them is zero. What I want to discuss is what is implied by the fact that so many university students of mathematics seem to think that for all real numbers and . (The misunderstanding is very widespread and well known: in the USA it’s often called the “Freshman’s Dream”.)
Before we do that, let me observe that ancient Greek children would have never made this mistake. For ancient Greeks real (positive) numbers meant lengths and their squares meant areas; their thinking was purely geometric and not algebraic as ours. Hence to compute they would have made the diagram as in the figure below, in which and are stippled, and it is clear that to get the whole area you have to add to the stippled squares the two rectangles of area .
Think about it differently. Wittgenstein once said that “solving a mathematical problem is like obeying a command”. As always with him, the more you think about what he says, the richer in meaning it becomes. For the Greek children the command was to create the diagram described in the Figure and then to follow the logic inherent in it. I claim something is wrong with the way many students interpret the command implied in .
Many people also seem to think that , which of course is wrong as you can see just by putting and remembering that . The same useful fact will also show you that in general (check with ).
So it seems that many students think that the command of , which is “add a and b and square the result” is equivalent to “square a and b and then add them”. In other words, they think that in “most” cases in mathematics (as exemplified above by the sin and the square root computations),
which would mean that they implicitly assume that all functions are linear. I thought that was the crux of the mistake, but now I think the situation is more complex. I have seen students write
which again is easily seen to be not true by taking , and quite frequently
I leave it to you to come up with a convincing counterexample.
My conclusion is that the “blanket assumption” is that and also that . [An interesting question is to find a function f for which this is true for all real a and b.]
Let us rephrase what seems to be happening. Confronted with the task of doing some operation on a and b and then applying the function f, many students think that this is equivalent to applying f to a and b and then doing the operation on the result.
Hence I think that means that many students (unconsciously) think that all operations in mathematics commute: that the order in which you perform the operations is not important.
But of course the order of operations is as important in mathematics as it is when you are putting on socks and shoes and it remains for me to marvel how it is possible that secondary school education does not explicitly disabuse you on this account.
(MG, with thanks to Drs. F. Goldman and I. Gibson of Glasgow University)
Here’s one way of looking at the problem — or perhaps just at part of it. When we use a mathematical idea, very often we’re using two related pictures of it at the same time: an informal version (sometimes called a concept image in the educational jargon) and a formal version (sometimes called a concept definition). For example, when you think of the concept of a continuous function, your informal picture probably involves drawing the graph without taking your pen off the paper, even when you know that the formal definition involves and and makes no mention of pen, paper, graph or drawing.
Almost all mathematicians do this: what distinguishes an expert mathematician is, first, that she has learned to refine her informal pictures so that they’re not misleading; and second, that she has learned how to convert an idea that was originally arrived at informally into a formal argument, and to check that the reasoning holds together.
When we meet a new piece of mathematics, we have not only to learn the definition but to start building up our image of what this definition “means”. (This is one reason that doing lots of examples is so important: it assembles a “gallery” of pictures from which we can build up our overall image of the concept.) Of course, being human beings we rarely if ever build up any mental image from scratch. Instead, we collect bits of previous images and try to reassemble them. (The great Scottish philosopher David Hume argued in An Enquiry Concerning Human Understanding that this is the only thing we can do: “all this creative power of the mind amounts to no more than the faculty of compounding, transposing, augmenting, or diminishing the materials afforded us by the senses and experience”.) In the case of the Freshman’s Dream, which ideas we reassemble depends on whether we’re thinking geometrically or algebraically.
As noted above, if we think geometrically we will never misunderstand what means. If we’re thinking algebraically, though, expressions like , , and all have a misleading family resemblance. Like me, you probably first encountered brackets in expressions like and , and spent many tedious hours in class multiplying them out (or, on a really thrilling day, factorising expressions to put the brackets back in). As a result of all this drill, even if we know formally that the brackets are there to specify the order of operations, our informal image of brackets still treats them as a signal to multiply out. I suspect it’s this urge to multiply out brackets at all costs that leads to the Freshman’s Dream — and to many other errors and abuses along similar lines.
As an aside, I don’t think this is the only place where this kind of “programming” trips people up. In school maths, we see a lot of quadratic expressions, and almost always what we’re being asked to do is to set them equal to zero and solve the resulting equations — usually by factorising, though sometimes by the quadratic formula. A result of this is that if I give a first-year class a question involving the expression then, no matter what I’m actually asking them to do, some people will go onto autopilot and write down the roots of the equation .
So how do we “deprogram” ourselves from this sort of habit — how do we make sure we’re not being tripped up by inadequate “images” of mathematical concepts? As I’ve said above, I think examples are important — but not just routine examples, because these rarely challenge our intuition. We need to look at the “pathological” cases, the ones that don’t seem to make sense at first, or perhaps don’t even seem to make sense even when we know we’ve got it right.
To break yourself of the Freshman’s Dream, I recommend doing plenty of problems involving logarithms, trig functions and factorials, until you’ve trained your brain out of multiplying out brackets. (Here’s a nice one to start with: simplify as much as possible. Dead easy as long as you remember the rules, but you might like to think of as many ways as you can to get it wrong…) Similarly, to break us of errors involving the definition of continuity, analysis lecturers love to present functions like Cantor’s staircase or the Weierstrass function.
So, unlike MG, I’m not so surprised that so many of us end up unconsciously believing that all operations commute; I think it follows almost inevitably from the fact that we’re educated in a mathematical culture which is dominated by algebra, and from the fact that we see so many routine examples of procedures and so few horrible pathological ones.
Here’s a nice intuition-buster to finish on, illustrating the fact that in 3D space, rotations don’t commute. Take a book and hold it with the front cover upward. Now, first rotate it clockwise through around the vertical axis, then through around the left-right horizontal axis. (The axes are defined relative to you, not to the book.) Note which way the book is facing. Now return it to its original position, and this time rotate it first around the horizontal axis and then around the vertical. If the book’s facing the same way as before, you’ve done something wrong. If it’s facing in a different direction and you find this unsurprising then congratulations — you’ve now really understood that rotations don’t commute!